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Orbital field angular momentum and torque balance

Physical scope

The conditional orbital angular momentum balance of the classical vacuum electromagnetic field about the fixed origin of the laboratory chart: the local identity linking the time derivative of the orbital angular momentum density and the divergence of its flux tensor to the torque density, and its fixed-volume form with the boundary angular flux retained, deduced from the momentum balance of the same framework. The density is orbital only; no spin or intrinsic angular momentum of matter is defined or claimed, and no empirical claim is made.

Statement

Assume the hypotheses of Local and fixed-volume momentum balance for smooth sources: the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches, clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities, the jointly $C^{1}$ fields of The electric and magnetic fields as fixed-time spatial vector fields, continuous sources, and the force density $f = ρ E + J \times B$ . Let $g$ , $σ$ , $Π = - σ$ , the orbital angular momentum density $ℓ = x \times g$ and its flux $x \times Π$ be those of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum relative to the origin fixed there. Then:

(a) [Local identity] On $Ω \times I$ ,

$\partial_{t} ℓ + div (x \times Π) = - x \times f,$

componentwise $\partial_{t} ℓ_{i} + \sum_{j = 1}^{3} \partial_{j} (x \times Π)_{i j} = - (x \times f)_{i}$ , where $(div (x \times Π))_{i} : = \sum_{j = 1}^{3} \partial_{j} (x \times Π)_{i j}$ and $(x \times f)_{i} = \sum_{j, k = 1}^{3} ϵ_{i j k} x_{j} f_{k}$ with the Levi-Civita symbol of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum, and $\partial_{t}$ is the partial time derivative at fixed position (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

(b) [Fixed-volume identity] Under the domain, time-interval and ${A C}_{ω}$ hypotheses of Local and fixed-volume momentum balance for smooth sources(b), and with $n$ the outward unit normal, for every $t \in (t_{0}, t_{1})$ and each $i$ ,

$\frac{d}{d t} \int_{Ω^{'}} ℓ_{i} d x = - \oint_{\partial Ω^{'}} (\sum_{j = 1}^{3} (x \times Π)_{i j} n_{j}) d S - \int_{Ω^{'}} (x \times f)_{i} d x,$

the closed-surface integral being the finite face sum of the divergence theorem used in that clause.

(c) [Scope] The identities depend on the origin fixed in Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum(e) through the explicit factor $x$ ; no field-only conservation follows by dropping the boundary flux: (b) retains both that flux and the integrated torque, which can cancel to give a zero field-angular-momentum rate. Conservation of field-plus-matter angular momentum requires explicit matter balances and their boundary fluxes. The hypotheses and exclusions of Local and fixed-volume momentum balance for smooth sources(c) are inherited. The density $ℓ$ is orbital, $ℓ = x \times g$ ; no spin or intrinsic angular momentum of matter is defined or claimed by this item. This is not an empirical claim.

Facts & Assumptions

Given: The hypotheses of Local and fixed-volume momentum balance for smooth sources; the objects $g, σ, Π, ℓ, x \times Π$ of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum; and, in clause (b), the admissible volume $Ω^{'}$ , the interval $(t_{0}, t_{1}) \subseteq I$ and ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Under the stated hypotheses the local momentum identity $\partial_{t} g + div Π = - f$ holds pointwise on $Ω \times I$ , with $g$ of class $C^{1}$ , $Π$ of class $C^{1}$ and symmetric, and $f$ continuous (Local and fixed-volume momentum balance for smooth sources).

[F2]

The objects of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum: $g$ , $σ$ and $Π = - σ$ are of class $C^{1}$ ; $Π$ is symmetric; the orbital angular momentum density is $ℓ = x \times g$ , componentwise $ℓ_{i} = \sum_{j, k = 1}^{3} ϵ_{i j k} x_{j} g_{k}$ , and its flux tensor is $(x \times Π)_{i ℓ} = \sum_{j, k = 1}^{3} ϵ_{i j k} x_{j} Π_{k ℓ}$ ; both depend on the fixed origin of the spatial chart. The cross product is that of The cross product in $R^{3}$ , and the Levi-Civita symbol is defined in Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum(d).

[F3]

Angular local identity of The quadratic stress identity and the local angular-balance identity: for open $V \subseteq R^{3}$ and open $I \subseteq R$ , let $Π : V \times I \to R^{3 \times 3}$ have continuous entries and spatial first partial derivatives, with $Π_{k j} = Π_{j k}$ . Let $g, f : V \times I \to R^{3}$ be continuous, with $\partial_{t} g$ existing componentwise and $\partial_{t} g + div Π = - f$ . Then $\partial_{t} (x \times g) + div (x \times Π) = - x \times f$ on $V \times I$ .

[F4]

Fixed-volume tools: for compact measurable $K$ with finite Lebesgue measure and $h, \partial_{t} h$ continuous on a neighbourhood of $K \times [a, b]$ , $t \mapsto \int_{K} h d λ_{n}$ is differentiable on $(a, b)$ with derivative $\int_{K} \partial_{t} h d λ_{n}$ (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface); if $Ω$ carries the finite piecewise $C^{1}$ presentation and $F \in C^{1} (\overline{Ω}; R^{n})$ , then $\int_{Ω} div F = \sum_{j} \int_{S_{j}} F \cdot ν_{j} d S$ with faces counted once off the edge set, under ${A C}_{ω}$ (Divergence for finite piecewise C1 presentations); and the boundary of such an $Ω$ is Lebesgue-null by the graph-coordinate Fubini and rigid-change-of-variables argument in Proof step 4.1 of that divergence theorem, followed by finite subadditivity over its faces; the integral of a nonnegative measurable function over a null set vanishes (A nonnegative integral over a null set vanishes), so signed integrals over $Ω$ and its closure agree.

Proof

technique · direct; apply the angular identity of the quadratic stress lemma to the momentum balance, and integrate the result over an admissible fixed volume with compact-parameter differentiation and the same divergence theorem

1.1givenF1F2F3algebra

Local identity: by [F1] the hypotheses of [F3] hold on $Ω \times I$ with $V = Ω$ : $g$ and the symmetric $Π$ are jointly $C^{1}$ , and $f$ is continuous. Applying [F3] gives $\partial_{t} (x \times g) + div (x \times Π) = - x \times f$ there, which is clause (a) with $ℓ = x \times g$ and the component conventions of [F2].

1.2givenF2F4algebra

Fixed-volume setup: let $Ω^{'}$ , $t_{0} < t_{1}$ be as in clause (b), fix $t \in (t_{0}, t_{1})$ and choose $a, b$ with $t_{0} < a < t < b < t_{1}$ . The closure $\overline{Ω^{'}}$ is compact and contained in the open $Ω$ , and each $ℓ_{i}$ together with $\partial_{t} ℓ_{i}$ is continuous on a neighbourhood of $\overline{Ω^{'}} \times [a, b]$ by [F2] and [F3]; the boundary is Lebesgue-null by the finite piecewise $C^{1}$ presentation and [F4], so integrals over $Ω^{'}$ and over its closure agree, and [F4] makes $t \mapsto \int_{Ω^{'}} ℓ_{i} (x, t) d x$ differentiable at $t$ with derivative $\int_{Ω^{'}} \partial_{t} ℓ_{i} (x, t) d x$ .

2.1step 1.1step 1.2F2F4algebra

Fixed-volume identity: by steps 1.2 and 1.1, $\frac{d}{d t} \int_{Ω^{'}} ℓ_{i} d x = \int_{Ω^{'}} \partial_{t} ℓ_{i} d x = - \int_{Ω^{'}} (x \times f)_{i} d x - \int_{Ω^{'}} (div (x \times Π))_{i} d x$ ; the fixed-time vector field $x \mapsto ((x \times Π)_{i 1}, (x \times Π)_{i 2}, (x \times Π)_{i 3}) (x, t)$ is $C^{1}$ on a neighbourhood of $\overline{Ω^{'}}$ by [F2], so the divergence theorem of [F4] gives $\int_{Ω^{'}} (div (x \times Π))_{i} d x = \oint_{\partial Ω^{'}} \sum_{j} (x \times Π)_{i j} n_{j} d S$ as a finite face sum, which is clause (b) under the stated ${A C}_{ω}$ .

3.1step 1.1step 2.1F1F2∎

Clause (c): the identity depends on the fixed origin through the explicit coordinate factor in $ℓ$ and $x \times Π$ (the shift law of [F2]); the boundary term is retained, and the hypotheses and exclusions of the momentum balance are inherited unchanged; no spinor or material intrinsic angular momentum enters the argument.

Remarks

Why symmetry of $Π$ is the whole content. Contracting the momentum balance with the position vector, the term produced by the derivative of $x$ is the antisymmetric contraction $ϵ_{i j k} Π_{k j}$ , which vanishes exactly because $Π$ is symmetric; the identity is therefore not a second dynamical law but a consequence of the momentum balance together with the symmetry recorded in the definition.
Origin dependence. Under a shift of origin the left and right sides change in the same way, since $x \mapsto x + a$ adds $a \times (\partial_{t} g + div Π)$ to the left-hand side and $- a \times f$ to the right and these are equal by the momentum balance; the displayed form nevertheless refers to the one fixed origin.

Orbital field angular momentum and torque balance

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources